Online Course Discussion Forum

MC III Number Theory 1.17,1.18,1.19

 
 
DingAlan的头像
MC III Number Theory 1.17,1.18,1.19
DingAlan - 2024年03月22日 Friday 21:02
 

May I have some ideas on how to start these? They were not explained in class. Thank you!

 
LinXilong的头像
Re: MC III Number Theory 1.17,1.18,1.19
LinXilong - 2024年03月22日 Friday 23:04
 
Can you please photo the original problem. 
DingAlan的头像
Re: MC III Number Theory 1.17,1.18,1.19
DingAlan - 2024年03月23日 Saturday 09:44
 


DingAlan的头像
Re: MC III Number Theory 1.17,1.18,1.19
DingAlan - 2024年03月23日 Saturday 09:48
 


LinXilong的头像
Re: MC III Number Theory 1.17,1.18,1.19
LinXilong - 2024年03月24日 Sunday 23:24
 
For 1.17, you can look at this https://artofproblemsolving.com/community/c6h1124783p5192873.

For 1.18, consider x^2 = (-x)^2.

For 1.19,write the left and side as n(n + 1)/2.

Good luck. 

DingAlan的头像
Re: MC III Number Theory 1.17,1.18,1.19
DingAlan - 2024年03月30日 Saturday 09:56
 

I couldn't follow the solutions on AoPS for 1.17, they were too confusing

I tried $m=2,4$ and I figured all sums I tested don't work 

but I can't figure out how to prove it for all m

maybe induction??


$\text{For 1.18}$,

$\text{I tried small values and they all showed that they were not complete residue systems.}$

I tried making $x^2$=$(-x)^2$ like you suggested but I don't really know what to do after that,


I'm figuring out what to do after making the left $\frac{n(n+1)}{2}$ for 1.19





LensmireJohn的头像
Re: MC III Number Theory 1.17,1.18,1.19
LensmireJohn - 2024年03月30日 Saturday 10:55
 

Hi Alan,

To start, it might be good to revisit 1.16 and the extra problem we called 16.5 in the live class. The relevant portions start around 1 Hour and 18 Minutes in the Week 2 Recording. Note: the key idea here is pairing a number a with (m-a). Note (m-a) can also be through of as -a if we're working mod m.

For 1.17, the extra problem 16.5 is actually one easy way to prove the result.

For 1.18, the pairing trick mentioned above is also the key.

For 1.19, it's probably easiest to think of two cases, once for n even and one for n odd. Both cases are similar but it can be easier to think of them separately. As a further hint, if n is even, try to prove (n/2) divides (1^k+...+n^k) and (n+1) divides (1^k+...+n^k). Again pairing will be helpful.

DingAlan的头像
Re: MC III Number Theory 1.17,1.18,1.19
DingAlan - 2024年04月9日 Tuesday 20:54
 

$\text{I figured out 1.17. Thanks for the help!}$


For 1.18, I couldn't find how to pair anything, maybe $k^2$ and $(m-k)^2$, so I used the fact that a complete residues system does not have repeated values. I couldn't solve it though.


I couldn't figure 1.19.